One thing I realized while writing and editing the previous article is the depth of the mismatch between notation for addition and multiplication (on the one hand) and exponentiation (on the other).

Students struggle with understanding the notation $$y = b^x$$, and one reason that’s been clear to me is that there’s no overt operator. It’s true that we don’t need to write the multiplication operator, but it’s an option. The use of the caret to indicate exponentiation, though, is not typically considered “standard” mathematics notation, but rather is still seen largely as the easiest way to type. TeX, calculators, and suchlike convert the caret into a superscript.

Beyond that, though, the basic functions students see prior to exponential and trigonometric functions have the variable appearing before the constant. We more typically write $$x + 3$$ than $$3 + x$$, for instance, although both are possible.

Radicals are in some sort of liminal realm between operators and functions: They feel more like a function, especially since square roots are often taught well before other roots. But they’re technically an operator, in that they come between their index (usually a parameter) and their argument (usually a variable).

Also, the unit on exponential functions is typically immediately after the one on rational exponents, where $$y = x^n$$ reinforces the familiar variable-then-parameter ordering.

So $$y = b^x$$ is “weird” on two levels: The operator is implied (through superscripting) rather than overt, and the variable is necessarily second, instead of preferentially first.

Since this is well-established notation, there’s no changing it any time soon, but it’s definitely something mathematics teachers should be aware of.

Clio Corvid